DR. PLUCKER ON A NEW GEOMETRY OE SPACE. 
753 
If the axis OY be turned round O till, in its new position OY', the angle Y'OX 
becoming 9, the plane ZOY' passes through the axis of the second complex, the last 
equation, by putting . 
<7=<r sm 9, 
r=r'-i-s'cos9, 
assumes the following form, 
o’ sin 9 = Jcr' + Jed cos 9. 
The axis of the second complex O' meets OZ in a point O', O'O being A. O' may be 
regarded as the origin of new coordinates, OY and OZ being replaced by OY" con- 
gruent with the axis of O', and by O'X" perpendicular to ZY" ; then the second com- 
plex O' will be represented by the equation 
g"=£'s", 
g" and s" being the new ray-coordinates and k' the constant of the complex. In order 
to make O'X" parallel to OX', it is to be turned round O' till, in its new position O'X'", 
the angle Y"'0'X" becomes 9. Accordingly, by putting 
g"=g'"sin 9, 
s"=r'"cos9+s'", 
the equation of the complex is transformed into the following, 
g'" sin 9=#V"' cos . 
Finally, by displacing the origin O' into O, g'" becomes g IV +Ar'", whence 
g'" sin 9=(&' cos 9+ A sin 9)r'"-|-&V. 
On omitting the accents, both complexes O and Q', referred to the same axes of 
coordinates OZ, OY', OX, the two last of which include an angle 9, are represented 
by the following equations, 
<rsin9=#r-4-/£ cos9.s, 1 
l (77) 
g sin 9=(#' cos 9+ A sin 9)r+&'s. j 
59. In order to determine the directrices of the congruency represented by the system 
of the last equations (77), the equations (65) and (66) may be transformed by putting 
A =k, B=£cos9, D=— sin 9, E=0, 
A'=&'cos9+ A sin9, B'=&', D'=0, E'= — sin 9 
into those following, 
0 =(z sin 9) 2 - [(&+£') cos 9+ A sin 9> sin 9+(M f sin 2 9-A£ sin 9- cos 9), . (78) 
A fy \ 2 (k'—k) cos — A sin y k 
